Energy spectrum of turbulent velocity pulsations in a wide range of wave numbers (renorm-group approach)

The problem of calculating the energy spectrum of turbulent velocity pulsations in the case of homogeneous isotropic and stationary turbulence is considered. The domain of turbulent energy production is treated as “a black box” on which boundary the spectral energy flux is given. It is assumed that the spectrum is formatted due to intermodal interactions being local in the wave-number space that leads to a cascade mechanism of energy transfer along the wave-number spectrum and the possibility of using the renormalization-group method related to the Markovian features of the process under consideration. The obtained formula for energy spectrum is valid in a wide wave-number range and at arbitrary values of fluid viscosity. It is shown that in functional formulation of the statistical theory of turbulence, the procedure of separating local intermodal interactions, which govern energy transfer (straining effect), and filtering out nonlocal interactions, which have no influence on energy transfer (sweeping effect), is directly described without providing additional arguments or conjectures commonly used in the renormalization-group analysis of turbulent spectra.

In order to find the shape of energy spectrum within the framework of the model of stationary homogeneous isotropic turbulence, the renormalization-group equations, which reflect the Markovian nature of the mechanism of energy transfer along the wavenumber spectrum, are used in addition to the dimensional considerations and the energy balance equation. For the spectrum, the formula depends on three parameters, namely, the wavenumber, which determines the upper boundary of the range of the turbulent energy production, the spectral flux through this boundary, and the fluid kinematic viscosity.

Summary The drag reducing effect of a surfactant additive from the alkylpolyglycosides family was investigated experimentally. In thepresent study, PIV measurements were done to investigate theturbulent flow in the streamwise-spanwise plane of a flume aty 1 580, in the buffer zone, where about 80% of the energy pro- Fig. 4 PDF’s of the u rms8 O U q for water —circles– and surfactant—squares–. u rms8 —x,z– ˜ −—u—x,z– A −u—x,z–‰– 2 ‰and the PDF is rep-resenting the spatial distribution of the rms values, normalizedby the flow-rate velocity U q .Fig. 5 Two-point—auto–correlation function R uu of the stream-wise velocity along the streamwise x and spanwise z coordi-nates, for water —circles and plus symbols– and for surfactant—square and triangular markers–, respectively.Table 1 Statistical properties of the Reynolds stress A −u 8 w 8 ‰ O U q2 .Statistics Maximum Minimum Average Std. dev. SkewnessWater 0.05 20.035 0.35310 23 0.0045 0.93Surfactant 0.03 20.035 0.35310 24 0.0032 0.003Table 2 Statistical properties of the turbulent energy production term

Magnetohydrodynamic (MHD) turbulence with cross-helicity excited by a large-scale source is considered. The source is a random external force contributing energy with a controlled level of cross-helicity. The proposed approach allows one to form and maintain high levels of cross-helicity without using an external constant magnetic field. The force is implemented in the software package TARANG. A series of numerical experiments with a constant energy flow and various levels of injection of cross-helicity have been carried out. The ratio of the cross-helicity to the total energy varied in the range from 0 to 0.6. The calculations were carried out at kinetic Reynolds number R = 2094 and at magnetic Prandtl number Prm= 1 on the grid 5123. An equal amount of energy per unit of time was introduced into velocity and magnetic induction fields. Regardless of the level of cross-helicity, the rate of dissipation of magnetic energy has always been higher than the intensity of injection. Spectral energy fluxes indicated a positive energy flux from the velocity field to the magnetic induction field. The high levels of cross helicity led to energy accumulation on a large scale and to changes in energy distribution over the scale. The spectral energy densities in the Elsasser variables z±showed a significant difference in the slopes of the z+and z-spectra. On the spectral flux z-, a region close to the inertia interval was detected. Qualitative agreement with the results obtained earlier using cascade models is shown. Varying the ratio of codirectional change of fields did not affect the integral and spectral characteristics of the flow. The results obtained allow us to use the proposed force for a detailed study of the evolution of homogeneous isotropic turbulence with a high level of cross-helicity and its dissipation.

The geographic character of the inverse cascade is analyzed based on the spectral kinetic energy flux calculated in the global ocean, using sea surface height (SSH) data from satellites, reanalysis data, and model outputs. It is shown that the strongest inverse cascade occurs mostly in high-energy eastward-flowing currents, such as the Antarctic Circumpolar Current (ACC), the Kuroshio Extension, and the Gulf Stream, which matches the global distribution pattern of the eddy kinetic energy (EKE). Hence, the eddy scales predicted by the local linear baroclinic instability and from the altimeter observation are mapped out and compared with the energy injection scale and the arrest-start scale of the inverse cascade, respectively. Generally, agrees well with in the midlatitude and high-latitude oceans, especially in the Northern Hemisphere. falls within the arrest ranges of the inverse cascade and is quite close to . Finally, the depth dependence and the anisotropy of the inverse kinetic energy cascade are also diagnosed in the global ocean. We have found that the strength of the inverse cascades decreases with increasing depth, but the global pattern of the strength is nearly invariable. Meanwhile, the variations in depth hardly affect the and . After considering the anisotropy in the spectral flux calculation, a possible inertial range for the zonal spectral kinetic energy flux is expected, where the cascade magnitude will keep a nearly constant negative value associated with the oceanic zonal jets.

Large-scale atmospheric turbulence is examined through the application of a wave-number frequency Fourier analysis of the velocity distributions on latitude circles. The primary advantage of the wave-number frequency energy spectra is that it permits analysis of the transient eddies in terms of their length scale and in terms of the speed and direction of their motion. Thus, the relative importance of retrogressing waves may be considered in the analysis. The kinetic energy spectra of the meridional component of velocity is found to contain a distinct band of energy for each wave number which shifts to higher negative frequencies as wave number increases from 4 through 10, at 40° N during winter season 1964. This indicates that the motion of these waves is predominantly from west to east. During the summer season, waves moving in the opposite direction carry a relatively large portion of the transient eddy kinetic energy. The terms of the component form of the kinetic energy equations in wave-number frequency space are evaluated with horizontal nondivergent velocities. The form of the equations used involves nonlinear interaction terms, which represent an inertial transfer process; ageostrophic terms, which represent the work done by the pressure and Coriolis forces; and the eddy frictional terms. Of the nonlinear interaction terms, the one which provides the largest positive or negative contribution to the spectral energy is the inertial term involving the longitudinal derivative. The largest interaction combination within each of these terms involves the spectral components of the kinetic energy production terms. This condition causes a positive contribution to the energy of waves moving from west to east and a negative contribution to waves moving from east to west. The contributions to the spectral energy through the interaction terms involving latitudinal derivatives are smaller with less consistency in sign. The interaction combinations representing the kinetic energy production in these terms have the same magnitude as several other interaction combinations. The contributions from the ageostrophic terms serve to balance the contributions from the interaction terms. The magnitude of these balancing contributions is from 2 to 5 times the magnitude of the spectral energy. DOI: 10.1111/j.2153-3490.1969.tb00484.x

Gupta, Lodato, and Scalo (JFM, 2017) have demonstrated the existence of an equilibrium spectral energy cascade in shock waves formed as a result of continued modal thermoacoustic amplification consistent with Kolmogorov’s theory for high-Reynolds-number hydrodynamic turbulence. In this work, we develop a rigorous theory of spectral energy cascade in an ensemble of nonlinear acoustic waves, which fully develop into randomly distributed shock waves resulting in acoustic wave turbulence. In analogy to hydrodynamic turbulence, the dynamics are shown very similar to the homogeneous isotropic turbulence in a box. To elucidate the energy dynamics, we derive mathematically exact energy corollary for second order nonlinear acoustics thus identifying the second-order energy norm for acoustics. For randomly initialized nonlinear waves, the mean energy in the domain decays with a −2/3 law in time due to coalescence of shock waves. In the spectral space, the energy corollary yields analytical expressions of spectral energy, spectral energy flux, and spectral energy dissipation. We derive the spectral energy scaling laws based on the Kolmogorov length scale which corresponds to the shock thickness in acoustic wave turbulence.

Plasma turbulence at scales of the order of the ion inertial length is mediated by several mechanisms, including linear wave damping, magnetic reconnection, the formation and dissipation of thin current sheets, and stochastic heating. It is now understood that the presence of localized coherent structures enhances the dissipation channels and the kinetic features of the plasma. However, no formal way of quantifying the relationship between scale-to-scale energy transfer and the presence of spatial structures has been presented so far. In the Letter we quantify such a relationship analyzing the results of a two-dimensional high-resolution Hall magnetohydrodynamic simulation. In particular, we employ the technique of space filtering to derive a spectral energy flux term which defines, in any point of the computational domain, the signed flux of spectral energy across a given wave number. The characterization of coherent structures is performed by means of a traditional two-dimensional wavelet transformation. By studying the correlation between the spectral energy flux and the wavelet amplitude, we demonstrate the strong relationship between scale-to-scale transfer and coherent structures. Furthermore, by conditioning one quantity with respect to the other, we are able for the first time to quantify the inhomogeneity of the turbulence cascade induced by topological structures in the magnetic field. Taking into account the low space-filling factor of coherent structures (i.e., they cover a small portion of space), it emerges that 80% of the spectral energy transfer (both in the direct and inverse cascade directions) is localized in about 50% of space, and 50% of the energy transfer is localized in only 25% of space.

Sea surface height measurements from satellites reveal the turbulent properties of the South Pacific Ocean surface geostrophic circulation, both supporting and challenging different aspects of geostrophic turbulence theory. A near-universal shape of the spectral kinetic energy flux is found and provides direct evidence of a source of kinetic energy near to or smaller than the deformation radius, consistent with linear instability theory. The spectral kinetic energy flux also reveals a net inverse cascade (i.e., a cascade to larger spatial scale), consistent with two-dimensional turbulence phenomenology. However, stratified geostrophic turbulence theory predicts an inverse cascade for the barotropic mode only; energy in the large-scale baroclinic modes undergoes a direct cascade toward the first-mode deformation scale. Thus if the surface geostrophic flow is predominately the first baroclinic mode, as expected for oceanic stratification profiles, then the observed inverse cascade contradicts geostrophic turbulence theory. The latter interpretation is argued for. Furthermore, and consistent with this interpretation, the inverse cascade arrest scale does not follow the Rhines arrest scale, as one would expect for the barotropic mode. A tentative revision of theory is proposed that would resolve the conflicts; however, further observations and idealized modeling experiments are needed to confirm, or refute, the revision. It is noted that no inertial range was found for the inverse cascade range of the spectrum, implying inertial range scaling, such as the established K−5/3 slope in the spectral kinetic energy density plot, is not applicable to the surface geostrophic flow.

The spectral kinetic-energy flux is an effective tool to analyze the kinetic-energy transfer across a range of length scales, also known as the kinetic-energy cascade. Three methods to calculate spectral energy fluxes have been widely used, hereafter the ΠA, ΠF, and ΠQ definitions. However, the relations among these three definitions have not been examined in detail. Moreover, the respective contribution of the normal strain and shear strain of the flow field to kinetic-energy cascade has not been estimated before. Here, we use the kinetic energy equations to rigorously compare these definitions. Then, we evaluate the spectral energy fluxes, as well as its decomposition into the normal-strain and shear-strain components for the North Pacific, using a dynamically consistent global eddying state estimate. We find that the data must be preprocessed first to obtain stable results from the ΠF and ΠQ definitions, but not for the ΠA definition. For the upper 500 m of the North Pacific, in the wavenumber ranges with inverse kinetic-energy cascade, both the normal and shear-strain flow components contribute significantly to the spectral energy fluxes. However, at high wavenumbers, the dominant contributor to forward kinetic-energy cascade is the normal-strain component. These results should help shed light on the underlying mechanism of inverse and forward energy cascades.

We have conducted statistical analysis of the characteristics of the velocity field pulsations, spectral energy flux, and energy dissipation rate in the framework of two types of shell models of developed turbulence. It is shown that the probability distribution functions of the spectral energy flux of the considered models differ fundamentally: in one model the energy flux fluctuates but remains mainly positive, with a very small fraction of negative values, while in the other model the range of fluctuations is much wider and the probability of negative values is comparable to the probability of positive values. For the same initial conditions, only the first-order statistical moment coincides. Qualitative coincidence is demonstrated by the distribution function of the energy dissipation rate. It is important that, in spite of such different statistical properties of the cascade energy flux, both models reproduce equally well the statistics of the high-order moments for the velocity field pulsations. This fact is noteworthy because the behavior of the high-order structural functions is believed to be determined by the probability distribution function of the energy dissipation rate and/or the spectral energy flux density.

A statistical model for describing the decay of developed isotropic turbulence of an incompressible fluid is proposed. The model uses the distribution function of the velocity pulsations introduced earlier by the authors on the basis of the principle of maximum randomness of the velocity field for a given spectral energy flux. The renormalization-group technique and e′ expansion are used to calculate the correlation functions of the velocity that occur in the equation of spectral energy balance. This leads to a closed equation for the dependence of the energy spectrum on the integral turbulence scalerc(t). In the inertial interval, this equation gives the Kolmogorov asymptotic spectrum, while for the time dependence ofrc(t) and the pulsation energye(t) it predicts the power lawsrc(t)∼t2/5 andr(t)∼t−6/5.

The influence of surface polaritons on spectral energy flux at different temperatures or distances to the surface of a plate was analyzed. The relations of the net heat flux between two parallel plates with the material type were also discussed. The results demonstrate that the effect of surface polaritons is dominated in the spectral energy flux at 300 K when the distance is decreased to 100 nm. In addition, the intensity of surface polaritons increases with the temperature. The net heat flux between two parallel plates has a nearly linear relation with the temperature and is closely related to the material type. It reaches up to 2.792×107 W/m2 between two SiC plates, approximately 3~6 orders of magnitude larger than that between two different materials. However, the net heat flux between SiC and Al is merely 2329.7 W/m2, even smaller than the result calculated by the classical stefan-boltzman law between two blackbodies.

A three-equation model of the turbulent transport of momentum and heat for simulating a circulation structure over the heat island in a stably stratified environment under nearly calm conditions is formulated. The turbulent kinetic energy E = (1/2)〈uiui〉 (where 〈 〉 indicates averaging), its spectral flux ϵ (dissipation), and the dispersion of turbulent fluctuations of temperature 〈θ2〉 are found from differential equations; thus the correct modeling of transport processes in the interface layer with the countergradient heat flux is assured. Turbulent fluxes of momentum, −〈uiuj〉, and heat, −〈uiθ〉, are determined from fully explicit “gradient diffusion” models. The E–ϵ–〈θ2〉 turbulence model minimizes difficulties in simulating the turbulent transport in a stably stratified environment and reduces efforts needed for the numerical implementation of the model. Numerical simulation of the turbulent structure of the penetrative convection over the heat island under conditions of stably stratified atmosphere demonstrates that the three-equation model is able to predict the circulation induced by the heat island, temperature distribution, root-mean-square fluctuations of the turbulent velocity and temperature fields, and spectral turbulent kinetic energy flux that are in good agreement with the experimental data and results of large-eddy simulations.

Abstract